Understanding Tipping Points in Climate and Sustainability

Understanding Tipping Points in Climate and Sustainability

Waterloo Women in Math Aug. 2014 Mary Lou Zeeman Bowdoin College

Thanks to: many friends & colleagues

MCRN

Educa9on

Curriculum materials! primary school to uni.!

Blog

Blog entries! every day in 2013!

GOOGLE: MPE 2013!

Mathematics and Climate Research Network!

Faculty, Postdocs!

Graduates! & Undergraduates! working together!

Online working groups!

Broadcast courses!

Meetings, etc.!

GOOGLE: MCRN Math!

Our Plan today • • • • •

Math and Sustainability AlternaEve Stable States Resilience Gradually Changing Environment Decision Support

Components of Sustainability Hum

an Fa

ctors

Natural Resources

Peter March

Math Gives Sustainability Coherence Hum

an Fa

ctors

Mathematical Sciences

Natural Resources

Peter March

Bringing Coherence to Sustainability Hum

an Fa

ctors

Geography

Natural Resources

Bringing Coherence to Sustainability Hum

an Fa

ctors

Literature “Flight Behavior” Barbara Kingsolver

Natural Resources

Extremely Interdisciplinary Hum

an Fa

ctors

Math

Natural Resources

Needs Interdisciplinary Courage

• In students – translate b/w disciplines • In educators

EducaEon: Empowerment by Hope

Intergovernmental Panel on Climate Change

Google IPCC 5th Assessment Report WG1 -­‐ Science Basis “Warming of the climate system is unequivocal, and since the 1950s, many of the observed changes are unprecedented over decades to millennia.”

Observed Change: Carbon Dioxide

Mauna Loa South Pole

IPCC AR5

Observed Change: Temperature

Twelfth Session of Working Group I

Approved Summary for Policymakers

Figure SPM.1 [FIGURE SUBJECT TO FINAL COPYEDIT]

Global average surface temperature change, 1850-­‐2012

Annual averages

Decadal averages

1850

2000

IPCC AR5

Observed Change: Ocean AcidificaEon

CO2

pH

IPCC AR5

Climate Change and IPCC IPCC

5th Assessment Report, WG1 – Science Basis “Human influence on the climate system is clear.”

Climate Change and IPCC IPCC

5th Assessment Report, WG1 – Science Basis “Human influence on the climate system is clear.”

But climate predicEon is sEll difficult… Why? • Complex System • Chaos • Uncertainty • Tipping Points

Climate Change and IPCC IPCC

5th Assessment Report, WG1 – Science Basis “Human influence on the climate system is clear.”

But climate predicEon is sEll difficult… Why? • Complex System • Chaos • Uncertainty • Tipping Points

Climate Change and IPCC IPCC

5th Assessment Report, WG1 – Science Basis “Human influence on the climate system is clear.”

But climate predicEon is sEll difficult… Why? • Complex System • Chaos • Uncertainty • Tipping Points

Similar Challenges Across Sustainability Hum

an Fa

ctors

Math

Natural Resources

But climate predicEon is sEll difficult… Why? • Complex System • Chaos • Uncertainty • Tipping Points

Our Plan today • • • • •

Math and Sustainability AlternaEve Stable States Resilience Gradually Changing Environment Decision Support

Research Theme: Model Hierarchies Insight Simple Dynamical

PredicEve ???

Complex ComputaEonal

What’s in between?

Research Theme: Model Hierarchies Insight Simple Dynamical

I’ll be talking here

PredicEve ???

Complex ComputaEonal

But you can be thinking here

When do insights persist thru’ to complex version?

Example: Earth’s Energy Balance

Kiehl, J. T. and Trenberth, K. E., 1997

Math and Climate

Hans Kaper & Hans Engler SIAM

John Marshall & Alan Plumb

Ray Pierrehumbert Ka-­‐Kit Tung

Example: Earth’s Energy Balance

To begin, we think of the homogeneous solidassphere, ignoring di↵erences in ignoring di↵erences in To Earth begin, as weathink of the Earth a homogeneous solid sphere, the atmosphere’s composition, di↵erences among continents and oceans, di↵erences the atmosphere’s composition, di↵erences among continents and oceans, di↵erences in topography, and allinother local features the system.

This sphere exposedThis to sphere is exposed to topography, and allof other local features of theissystem. radiation from the Sun,radiation which is from is essentially point is source at infinity; see Figure the Sun,awhich is essentially a point source2.4. at infinity; see Figure 2.4.

Energy In

Balances

Energy Out

Figure 2.4: Incoming Figure sunlight (shortwave and outgoing heat (longwave 2.4: Incomingradiation) sunlight (shortwave radiation) and outgoing heat (longwave radiation). radiation).

17

17

Example: Earth’s Energy Balance

To begin, we think of the homogeneous solidassphere, ignoring di↵erences in ignoring di↵erences in To Earth begin, as weathink of the Earth a homogeneous solid sphere, the atmosphere’s composition, di↵erences among continents and oceans, di↵erences the atmosphere’s composition, di↵erences among continents and oceans, di↵erences in topography, and allinother local features the system.

This sphere exposedThis to sphere is exposed to topography, and allof other local features of theissystem. radiation from the Sun,radiation which is from is essentially point is source at infinity; see Figure the Sun,awhich is essentially a point source2.4. at infinity; see Figure 2.4.

Energy In

Balances

Energy Out

Figure 2.4: Incoming Figure sunlight (shortwave and outgoing heat (longwave 2.4: Incomingradiation) sunlight (shortwave radiation) and outgoing heat (longwave Depends o n A lbedo radiation). radiation). > 0.8 for ice and snow 17 17 < 0.2 for blue ocean

Albedo = FracEon reflected

Example: Earth’s Energy Balance

To begin, we think of the homogeneous solidassphere, ignoring di↵erences in ignoring di↵erences in To Earth begin, as weathink of the Earth a homogeneous solid sphere, the atmosphere’s composition, di↵erences among continents and oceans, di↵erences the atmosphere’s composition, di↵erences among continents and oceans, di↵erences in topography, and allinother local features the system.

This sphere exposedThis to sphere is exposed to topography, and allof other local features of theissystem. radiation from the Sun,radiation which is from is essentially point is source at infinity; see Figure the Sun,awhich is essentially a point source2.4. at infinity; see Figure 2.4.

Energy In

Balances

Energy Out

Figure 2.4: Incoming Figure sunlight (shortwave and outgoing heat (longwave 2.4: Incomingradiation) sunlight (shortwave radiation) and outgoing heat (longwave Depends o n A lbedo radiation). radiation). > 0.8 for ice and snow 17 17 < 0.2 for blue ocean

Energy In, extreme cases Temp very low => ice covered earth => Energy in is low

Example: Earth’s Energy Balance

To begin, we think of the homogeneous solidassphere, ignoring di↵erences in ignoring di↵erences in To Earth begin, as weathink of the Earth a homogeneous solid sphere, the atmosphere’s composition, di↵erences among continents and oceans, di↵erences the atmosphere’s composition, di↵erences among continents and oceans, di↵erences in topography, and allinother local features the system.

This sphere exposedThis to sphere is exposed to topography, and allof other local features of theissystem. radiation from the Sun,radiation which is from is essentially point is source at infinity; see Figure the Sun,awhich is essentially a point source2.4. at infinity; see Figure 2.4.

Energy In

Balances

Energy Out

Figure 2.4: Incoming Figure sunlight (shortwave and outgoing heat (longwave 2.4: Incomingradiation) sunlight (shortwave radiation) and outgoing heat (longwave Depends o n A lbedo radiation). radiation). > 0.8 for ice and snow 17 17 < 0.2 for blue ocean

Energy In, extreme cases Temp very low => ice covered earth => Energy in is low Temp very high => ice free earth => Energy in is high

Example: Earth’s Energy Balance

To begin, we think of the homogeneous solidassphere, ignoring di↵erences in ignoring di↵erences in To Earth begin, as weathink of the Earth a homogeneous solid sphere, the atmosphere’s composition, di↵erences among continents and oceans, di↵erences the atmosphere’s composition, di↵erences among continents and oceans, di↵erences in topography, and allinother local features the system.

This sphere exposedThis to sphere is exposed to topography, and allof other local features of theissystem. radiation from the Sun,radiation which is from is essentially point is source at infinity; see Figure the Sun,awhich is essentially a point source2.4. at infinity; see Figure 2.4.

Energy

Energy In

Balances

Energy Out

Figure 2.4: Incoming Figure sunlight (shortwave and outgoing heat (longwave 2.4: Incomingradiation) sunlight (shortwave radiation) and outgoing heat (longwave radiation). radiation).

Snowball

Ice free 17

17

Energy In, extreme cases Temp very low => Energy in is low Temp very high => Energy in is high

Global average temperature, T

Example: Earth’s Energy Balance

To begin, we think of the homogeneous solidassphere, ignoring di↵erences in ignoring di↵erences in To Earth begin, as weathink of the Earth a homogeneous solid sphere, the atmosphere’s composition, di↵erences among continents and oceans, di↵erences the atmosphere’s composition, di↵erences among continents and oceans, di↵erences in topography, and allinother local features the system.

This sphere exposedThis to sphere is exposed to topography, and allof other local features of theissystem. radiation from the Sun,radiation which is from is essentially point is source at infinity; see Figure the Sun,awhich is essentially a point source2.4. at infinity; see Figure 2.4.

Energy

Energy In

Energy Out

Balances

Energy radiated out:

Figure 2.4: Incoming Figure sunlight (shortwave and outgoing heat (longwave 2.4: Incomingradiation) sunlight (shortwave radiation) and outgoing heat (longwave radiation). radiation).

17

Global average temperature, T

17

ε σT4

Blackbody Stefan-­‐ Boltzmann RadiaEon constant

Example: Earth’s Energy Balance

To begin, we think of the homogeneous solidassphere, ignoring di↵erences in ignoring di↵erences in To Earth begin, as weathink of the Earth a homogeneous solid sphere, the atmosphere’s composition, di↵erences among continents and oceans, di↵erences the atmosphere’s composition, di↵erences among continents and oceans, di↵erences in topography, and allinother local features the system.

This sphere exposedThis to sphere is exposed to topography, and allof other local features of theissystem. radiation from the Sun,radiation which is from is essentially point is source at infinity; see Figure the Sun,awhich is essentially a point source2.4. at infinity; see Figure 2.4.

Energy

Energy In

Balances

Energy Out

Energy radiated out:

Figure 2.4: Incoming Figure sunlight (shortwave and outgoing heat (longwave 2.4: Incomingradiation) sunlight (shortwave radiation) and outgoing heat (longwave radiation). radiation).

17

Global average temperature, T

17

ε σT4

Atmospheric greenhouse gases, 0<ε<1

Blackbody RadiaEon

Example: Earth’s Energy Balance

To begin, we think of the homogeneous solidassphere, ignoring di↵erences in ignoring di↵erences in To Earth begin, as weathink of the Earth a homogeneous solid sphere, the atmosphere’s composition, di↵erences among continents and oceans, di↵erences the atmosphere’s composition, di↵erences among continents and oceans, di↵erences in topography, and allinother local features the system.

This sphere exposedThis to sphere is exposed to topography, and allof other local features of theissystem. radiation from the Sun,radiation which is from is essentially point is source at infinity; see Figure the Sun,awhich is essentially a point source2.4. at infinity; see Figure 2.4.

Energy Out

Balances

Figure 2.4: Incoming Figure sunlight (shortwave and outgoing heat (longwave 2.4: Incomingradiation) sunlight (shortwave radiation) and outgoing heat (longwave radiation). radiation).

17

Global average temperature, T

Energy

Energy

Energy In

εσT4

17

Global average temperature, T

Example: Earth’s Energy Balance Earth’s global average temperature is driven by energy imbalance

Energy

dT = C(energy in -­‐ energy out) dt energy out, εσT4

energy in

Global average temperature, T

Example: Earth’s Energy Balance Earth’s global average temperature is driven by energy imbalance dT = C(energy in -­‐ energy out) dt Energy

energy out, εσT4

a What happens when T=a?

energy in

T

Example: Earth’s Energy Balance Earth’s global average temperature is driven by energy imbalance dT = C(energy in -­‐ energy out) dt Energy

energy out, εσT4

a dT = 0 dt So T is in equilibrium At a:

energy in

T

Example: Earth’s Energy Balance Earth’s global average temperature is driven by energy imbalance dT = C(energy in -­‐ energy out) dt Energy

energy out, εσT4 energy in

a

b

c

dT At a, b and c: = 0 dt So T is in equilibrium

T

Example: Earth’s Energy Balance Earth’s global average temperature is driven by energy imbalance dT = C(energy in -­‐ energy out) dt Energy

energy out, εσT4 energy in

a

b

c

What if T is between b & c?

T

Example: Earth’s Energy Balance Earth’s global average temperature is driven by energy imbalance dT = C(energy in -­‐ energy out) dt Energy

energy out, εσT4 energy in

a

b Between b & c: So T is increasing

c dT > 0 dt

T

Example: Earth’s Energy Balance Earth’s global average temperature is driven by energy imbalance dT = C(energy in -­‐ energy out) dt Energy

energy out, εσT4 energy in

a

b

c

T

What if T is above c?

Example: Earth’s Energy Balance Earth’s global average temperature is driven by energy imbalance dT = C(energy in -­‐ energy out) dt Energy

energy out, εσT4 energy in

a

b

c

T

dT < 0 dt So T is decreasing Above c:

Example: Earth’s Energy Balance Earth’s global average temperature is driven by energy imbalance dT = C(energy in -­‐ energy out) dt Energy

energy out, εσT4 energy in

a

b

c

T

Example: Earth’s Energy Balance dT = C(energy in -­‐ energy out) dt

Energy

energy out, εσT4 energy in

a

b

c

T

a

b

c

T

Example: Earth’s Energy Balance dT = C(energy in -­‐ energy out) dt

Phase line for T

Energy

energy out, εσT4 energy in

a

c

b

b

c

T

a

Example: Earth’s Energy Balance dT = C(energy in -­‐ energy out) dt

Phase line for T

Energy

energy out, εσT4 energy in

a

Earth with c Ice caps b

b

c

T

Snowball a Earth

AlternaEve Stable States Examples • Earth’s energy balance

Phase line Temp.

Earth with c Ice caps

b

Snowball Earth a

AlternaEve Stable States Examples • Earth’s energy balance Geological evidence: 600 million years earth was a snowball.

QuesEon: How did we ever get out of that “stable” state?

Phase line Temp.

Earth with c Ice caps

b

Snowball Earth a

AlternaEve Stable States Examples • Earth’s energy balance Geological evidence: 600 million years earth was a snowball.

QuesEon: How did we ever get out of that “stable” state?

First: Let’s see some more examples

Phase line Temp.

Earth with c Ice caps

b

Snowball Earth a

AlternaEve Stable States in Shallow Lakes Clear Lake: • Lots of vegetaEon • Biodiversity among plankton, fish and birds • Sediment anchored.

Turbid Lake: • Algae dominate, block light.

• Lisle or no vegetaEon. • Loss of biodiversity • Waves and bosom feeders sEr up sediment.

Ecology of Shallow Lakes, Marten Scheffer, Springer, 2005

AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake

Phase line VegetaEon Clear c High vegetaEon

b

Sudden dramaEc change can, and does, happen.

Turbid a Low vegetaEon

Example: Florida Everglades

Sawgrass

Casails

AlternaEve Stable States in Florida Everglades • Sawgrass and casails compete • Casails are taking over

AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake • Everglades

Phase line Angle

Sawgrass c

b

Sudden dramaEc change can, and does, happen.

Casails a

AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake • Everglades • Coral reefs

Phase line Angle

Coral dominated c

b

Sudden dramaEc change can, and does, happen.

Algal dominated a

AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake • Everglades • Coral reefs • Fisheries

Phase line Biomass

Fish stock c

b

Sudden dramaEc change can, and does, happen.

No fish a

Somatotrophe Growth hormone cell in the pituitary.

AlternaEve stable states • Quiescent • Tonic firing • Tips back and forth

nt

!

hA, of of ihe o-

oaded from jn.physiology.org on July 3, 2008

fel ll he

BursEng in Excitable Cells

* Detail

AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake • Everglades • Coral reefs • Fisheries • Neuron

Phase line Membrane voltage Tonic firing c

b

Sudden dramaEc change can, and does, happen.

Quiescent a

AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake • Everglades • Coral reefs • Fisheries • Neuron • Neural network

Phase line Overall network acEvity High acEvity c

b

Sudden dramaEc change can, and does, happen.

Low acEvity a

AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake • Everglades • Coral reefs • Fisheries • Neuron, Neural network • Kayak

Phase line Angle

Upright c

b

Sudden dramaEc change can, and does, happen.

Upside down a

AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake • Everglades • Coral reefs • Fisheries • Neuron, Neural network • Kayak • Stress axis

Phase line Behavior

Fight c

b

Sudden dramaEc change can, and does, happen.

Flight a

AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake • Everglades • Coral reefs • Fisheries • Neuron, Neural network • Kayak • Stress axis • Mood

Sudden dramaEc change can, and does, happen.

Phase line Mood

Manic c

b

Depressed a

Research Theme: Model Hierarchies Insight Simple Dynamical

I’ll be talking here

PredicEve ???

Complex ComputaEonal

But you can be thinking here

When do insights persist thru’ to complex version?

Sudden dramaEc change can, and does, happen.

Our Plan today • • • • •

Math and Sustainability AlternaEve Stable States Resilience Gradually Changing Environment Decision Support

AlternaEve Stable States Examples • Earth’s energy balance • Shallow lake • Everglades • Coral reefs • Fisheries Policy/management quesEons: • Neuron, Neural network Are we at risk of a regime shiv?

• Kayak Is oaur state ‘resilient’ enough?

• Stress xis

• Mood How do we avoid a regime shiv?

How do we maintain enough resilience?

How do we induce a regime shiv?

How do we reduce resilience?

Phase line Mood

Manic c

b

Depressed a

Resilience… of what? to what?

Resilience: How much disturbance the system can withstand.

Phase line

c

b

a

Resilience… of what? to what?

Resilience: How much disturbance the system can withstand.

Phase line

Suppose the system is at state c c

b

a

Resilience… of what? to what?

Resilience: How much disturbance the system can withstand.

Phase line

Suppose the system is at state c c Disturb the system to here What happens? b

a

Resilience… of what? to what?

Resilience: How much disturbance the system can withstand.

Phase line

Suppose the system is at state c c The system returns to state c. It “recovers”

Disturb the system to here What happens? b

a

Resilience… of what? to what?

Resilience: How much disturbance the system can withstand.

Phase line

Suppose the system is at state c c

Now disturb the system to here What happens?

b

a

Resilience… of what? to what?

Resilience: How much disturbance the system can withstand.

Phase line

Suppose the system is at state c c

Now disturb the system to here What happens?

b

Regime shiv to state a. a

System does not recover it’s old state

Resilience… of what? to what?

Resilience: How much disturbance the system can withstand.

Phase line

Suppose the system is at state c c

Now disturb the system to here What happens?

b

b acts as a threshold Regime shiv to state a.

a

System does not recover it’s old state

Example: Shallow Lake Resilience: How much disturbance the system can withstand.

VegetaEon

c

Clear High vegetaEon,

b

a

Turbid Low vegetaEon,

Example: Shallow Lake Resilience: How much disturbance the system can withstand.

VegetaEon

Suppose the lake is clear and vegetated c

Clear High vegetaEon,

b

a

Turbid Low vegetaEon,

Example: Shallow Lake Resilience: How much disturbance the system can withstand.

VegetaEon

Suppose the lake is clear and vegetated c

If too much algae grow (e.g. from high nutrient loading)

Clear High vegetaEon,

b

a

Turbid Low vegetaEon,

Example: Shallow Lake Resilience: How much disturbance the system can withstand.

VegetaEon

Suppose the lake is clear and vegetated c

If too much algae grow (e.g. from high nutrient loading)

Lake shivs to turbid state

Clear High vegetaEon,

b

a

Turbid Low vegetaEon,

Example: Shallow Lake Resilience: How much disturbance the system can withstand.

VegetaEon

c

Clear High vegetaEon,

b

Suppose the lake is turbid a

Turbid Low vegetaEon,

Example: Shallow Lake Resilience: How much disturbance the system can withstand.

VegetaEon

c

If sediment disturbance is reduced (e.g. by removing some bosom feeder fish)

Clear High vegetaEon,

b

Suppose the lake is turbid a

Turbid Low vegetaEon,

Example: Shallow Lake Resilience: How much disturbance the system can withstand.

VegetaEon

Lake self-­‐recovers to clear state c If sediment disturbance is reduced (e.g. by removing some bosom feeder fish)

Clear High vegetaEon,

b

Suppose the lake is turbid a

Turbid Low vegetaEon,

QuanEfying Resilience Resilience: How much disturbance the system can withstand.

Resilience: Radius of basin of asracEon of the stable state

c

Basin of asracEon of stable state c

b

a

Basin of asracEon of stable state a

QuanEfying Resilience Resilience: How much disturbance the system can withstand.

Resilience: Radius of basin of asracEon of the stable state

c

QuesEons: What’s the distribuEon of disturbance/noise/shocks?

Basin of asracEon of stable state c

b

a

Basin of asracEon of stable state a

QuanEfying Resilience Resilience: How much disturbance the system can withstand.

Resilience: Radius of basin of asracEon of the stable state

c

QuesEons: What’s the distribuEon of disturbance/noise/shocks?

Basin of asracEon of stable state c

b

How does disturbance accumulaEon interact with dynamics? a

Basin of asracEon of stable state a

QuanEfying Resilience Resilience: How much disturbance the system can withstand.

Resilience: Radius of basin of asracEon of the stable state

QuesEon: Is the basin ‘large’ enough to contain accumulated disturbance?

c

QuesEons: What’s the distribuEon of disturbance/noise/shocks?

Basin of asracEon of stable state c

b

How does disturbance accumulaEon interact with dynamics? a

Basin of asracEon of stable state a

Research Theme: Model Hierarchies Insight Simple Dynamical

I’ll be talking here

PredicEve ???

Complex ComputaEonal

But you can be thinking here

Size and strength of basins of asracEon relaEve to Noise characterisEcs & accumulaEon behavior?

Our Plan today • • • • •

Math and Sustainability AlternaEve Stable States Resilience Gradually Changing Environment Decision Support

Channeling Christopher Zeeman Google: Sir Christopher Zeeman Christmas Lectures

EC Zeeman’s Framework

Discrete ConEnuous

BEHAVIOR

Discrete

THINGS

Dice Symmetries DISCRETE BOX

ConEnuous Tipping Points CriEcal Thresholds Phase TransiEons

Finite Probability Finite Groups

BifurcaEon Theory Catastrophe Theory

Planets PopulaEons

Waves ElasEcity

Ordinary DifferenEal Eq ParEal DifferenEal Eq

EC Zeeman’s Framework

Discrete ConEnuous

BEHAVIOR

Discrete

THINGS

Dice Symmetries DISCRETE BOX

ConEnuous Tipping Points CriEcal Thresholds Phase TransiEons

Finite Probability Finite Groups

BifurcaEon Theory Catastrophe Theory

Planets PopulaEons

Waves ElasEcity

TIME BOX Ordinary DifferenEal Eq ParEal DifferenEal Eq

EC Zeeman’s Framework

Discrete ConEnuous

BEHAVIOR

Discrete

THINGS

Dice Symmetries DISCRETE BOX

ConEnuous Tipping Points CriEcal Thresholds Phase TransiEons

Finite Probability Finite Groups

BifurcaEon Theory Catastrophe Theory

Planets PopulaEons

Waves ElasEcity

TIME BOX CONTINUOUS BOX Ordinary DifferenEal Eq ParEal DifferenEal Eq

EC Zeeman’s Framework

Discrete ConEnuous

BEHAVIOR

Discrete

THINGS

Dice Symmetries DISCRETE BOX

ConEnuous Tipping Points CriEcal Thresholds Phase TransiEons

Finite Probability Finite Groups

BifurcaEon Theory Catastrophe Theory

Planets PopulaEons

Waves ElasEcity

TIME BOX CONTINUOUS BOX Ordinary DifferenEal Eq ParEal DifferenEal Eq

EC Zeeman’s Framework

Discrete ConEnuous

BEHAVIOR

Discrete

THINGS

Finite Probability Finite Groups

ConEnuous Tipping Points CriEcal Thresholds Phase TransiEons PANDORA’S BOX BifurcaEon Theory Catastrophe Theory

Planets PopulaEons

Waves ElasEcity

Dice Symmetries DISCRETE BOX

TIME BOX CONTINUOUS BOX Ordinary DifferenEal Eq ParEal DifferenEal Eq

Tipping point as fold bifurcaEon

Behavior

1-­‐d slowly varying parameter

Research Challenge: generalize thru’ model hierarchies

Example: Earth’s Energy Balance

To begin, we think of the homogeneous solidassphere, ignoring di↵erences in ignoring di↵erences in To Earth begin, as weathink of the Earth a homogeneous solid sphere, the atmosphere’s composition, di↵erences among continents and oceans, di↵erences the atmosphere’s composition, di↵erences among continents and oceans, di↵erences in topography, and allinother local features the system.

This sphere exposedThis to sphere is exposed to topography, and allof other local features of theissystem. radiation from the Sun,radiation which is from is essentially point is source at infinity; see Figure the Sun,awhich is essentially a point source2.4. at infinity; see Figure 2.4.

Energy Out

Balances

Figure 2.4: Incoming Figure sunlight (shortwave and outgoing heat (longwave 2.4: Incomingradiation) sunlight (shortwave radiation) and outgoing heat (longwave radiation). radiation).

17

Energy

Energy

Energy In

17

εσT4

ε represents

greenhouse gases

Global average temperature, T

Global average temperature, T

Example: Earth’s Energy Balance dT = C(energy In -­‐ energy out) dt

Phase line for T

Energy

energy out, εσT4 energy in

a

c

b

b

c

Let’s increase ε (lower greenhouse gases)

T

a

Example: Earth’s Energy Balance dT = C(energy In -­‐ energy out) dt

Phase line for T

Energy

energy out, εσT4 energy in

a

c b

b

c

Increase ε (lower greenhouse gases)

T

a

Example: Earth’s Energy Balance dT = C(energy In -­‐ energy out) dt

Phase line for T

Energy

energy out, εσT4 energy in

a

c

Increase ε (lower greenhouse gases)

T

a

Example: Earth’s Energy Balance dT = C(energy In -­‐ energy out) dt

Phase line for T

Energy

energy out, εσT4

a

Increase ε (lower greenhouse gases)

energy in

T

Snowball Earth a

Example: Earth’s Energy Balance dT = C(energy In -­‐ energy out) dt

Phase line for T

Energy

energy out, εσT4 energy in

a

c

b

b

c

T

Now let’s decrease ε (increase greenhouse gases)

a

Example: Earth’s Energy Balance dT = C(energy In -­‐ energy out) dt

Phase line for T

Energy

energy out, εσT4 energy in

a

c

b

b

c

Decrease ε (increase greenhouse gases)

a

T

Example: Earth’s Energy Balance

Energy

dT = C(energy In -­‐ energy out) dt

Phase line for T Earth with ice caps c or no ice

energy out, εσT4

energy in

c

Decrease ε (increase greenhouse gases)

T

Global average temp, T

Stack up all the phase lines Stable States

Low

High

Greenhouse gas conc.

Example: Earth’s Energy Balance T

Snowball Earth Low

Greenhouse gas conc.

Example: Earth’s Energy Balance T

Earth with ice caps or no ice

High

Greenhouse gas conc.

Example: Earth’s Energy Balance T

Medium

Greenhouse gas conc.

Remember this quesEon? Phase line

Geological evidence: 600 million years earth was a snowball.

QuesEon: How did we ever get out of that stable state?

Temp.

Earth with c Ice caps

b

Snowball Earth a

Remember this quesEon? Phase line

Geological evidence: 600 million years earth was a snowball.

QuesEon: How did we ever get out of that stable state?

Watch what happens to the resilience (basin of asracEon)

Temp.

Earth with c Ice caps

b

Snowball Earth a

Change Environment T

Snowball Low

Medium

High

Greenhouse gas conc.

Gradually increase greenhouse gas concentraEon (volcanoes)

Change Environment T

Snowball Low

Medium

High

Greenhouse gas conc.

Gradually increase greenhouse gas concentraEon (volcanoes)

Change Environment T

Snowball Low

Medium

High

Greenhouse gas conc.

Gradually increase greenhouse gas concentraEon (volcanoes)

Change Environment T

Snowball Low

Medium

High

Greenhouse gas conc.

Gradually increase greenhouse gas concentraEon (volcanoes)

Change Environment T

Earth with ice caps or no ice

Low

Medium

High

Greenhouse gas conc.

Gradually increase greenhouse gas concentraEon (volcanoes)

Change Environment T

Earth with ice caps or no ice

Low

Medium

High

Greenhouse gas conc.

Now decrease greenhouse gas concentraEon (rocks and oceans)

Irreversibility as hysteresis T

Earth with ice caps or no ice

Low

Medium

High

Greenhouse gas conc.

PaleoClimate Record, 70M years Temperature

Million years ago

Now

Zachos

70 mill yrs ago

Irreversibility as hysteresis T

Earth with ice caps or no ice Last 70 mill yrs

Low

Medium

High

Greenhouse gas conc.

PaleoClimate Record, 70M years Temperature

Million years ago

Now

Zachos

Lots of interesEng abrupt events & Epping points within here, too -­‐ Requires more detailed models 70 mill yrs ago

Revisit Our Examples Examples • Earth’s energy balance • Shallow lake

Phase line VegetaEon Clear c High vegetaEon

b

Turbid a Low vegetaEon

Nutrient loading in shallow lake

Turbidity

Clear

Turbid High

Medium

Low

Nutrient Loading

Gradually Changing Environment Examples • Earth’s energy balance • Shallow lake • Everglades • Coral reefs

Nutrient run-­‐off from agriculture shrinks basin of asracEon

Then shock triggers regime change

Phase line

Clear Sawgrass c Coral dominated

b

Turbid Casails a Algal dominated

Our Plan today • • • • •

Math and Sustainability AlternaEve Stable States Resilience Gradually Changing Environment Decision Support

Can Critical Transitions be predicted ?

Even if we do not understand the system ?

Marten Scheffer

Research Theme: Model Hierarchies Insight Simple Dynamical

Spot a pasern here

PredicEve ???

Complex ComputaEonal

Test it here…

Resilience in Changing Environment Resilience Ability of system to withstand perturbaEon Size/strength of basin of asracEon of stable state

Equilibrium states of system

Slowly changing environment

Resilience in Changing Environment Resilience Ability of system to withstand perturbaEon Size/strength of basin of asracEon of stable state Resilience shrinks to zero as we approach Epping point Exploit that structure!

Equilibrium states of system

Slowly changing environment

Early Warning Signs?

Scheffer et al, Nature 2009 & Science 2012 How does system respond to stochasEc perturb.?

1.5

1

0.5

0

−0.5

−1

−1.5

−2 0

50

100

150

200

250

300

Early Warning Signs?

Scheffer et al, Nature 2009 & Science 2012 How does system respond to stochasEc perturb.?

2

1.5

1.5

1 1

0.5 0.5

0

0 −0.5

−0.5

−1

−1 −1.5

−1.5

−2 −2.5

0

50

100

150

200

250

300

−2 0

50

100

150

200

250

300

Early Warning Signs?

How does system respond to stochasEc perturb.?

CriEcal slowing: as system approaches bifurcaEon, rate of asracEon to equilibrium approaches 0 2 1.5 1

PerturbaEons die down more slowly

0.5

0

• Increased system ‘memory’ • Increased autocorrelaEon • Increased variance

−0.5 −1 −1.5 −2 −2.5

0

50

100

150

200

250

300

−0.6 −1.6 −1.65

−0.7

−1.7

−0.8

−1.75 −1.8

−0.9

−1.85 −1.9

−1

−1.95

−1.1 −2 −2.05

−1.2

−2.1 45

50

55

60

65

70

75

225

230

235

240

245

250

255

PaleoClimate Record, 70M years Temperature

Million years ago

Now

Zachos

Lots of interesEng abrupt events & Epping points within here, too

70 mill yrs ago

Slowing down precedes ancient shifts shifts Critical slowing down announced 8 climate abrupt climate

Dakos et al PNAS 2008

Research Theme: Model Hierarchies Insight Simple Dynamical

Spot a pasern here

PredicEve ???

Complex ComputaEonal

Test it here…

Shallow lakes Financial markets Power grid Psychology

Research Theme: Model Hierarchies Insight Simple Dynamical

Spot a pasern here

PredicEve ???

Complex ComputaEonal

Test it here…

InteresEng how oven it works. Inspiring new research quesEons.

Policy Decision Context We have some control over our gradually changing environment.

E.g. Land Use, Water, Fisheries, Urban planning, TransportaEon…

Equilibrium states of system

control of environment

Policy Decision Context We have some control over our gradually changing environment.

E.g. Land Use, Water, Fisheries, Urban planning, TransportaEon…

Policy goal: Manage for resilience Modeling goal: quanEfy effect of policy opEons on resilience Equilibrium states of system

control of environment

Policy Decision Context

What if decision maker has two dimensions of control opEons?

Equilibrium states of system

1-­‐d control of environment

Channeling EC Zeeman again

ECZ and his catastrophe machine

Channeling EC Zeeman again Generic picture for systems with minimizing dynamics

2-­‐d control of environment

Channeling EC Zeeman again Generic picture for systems with minimizing dynamics

No

r 2-­‐d control of o t c a f rmal environment

Channeling EC Zeeman again Generic picture for systems with minimizing dynamics

No

r 2-­‐d control of o t c a f rmal environment

Channeling EC Zeeman again Generic picture for systems with minimizing dynamics Splizng Factors:

Strength of Feedback

Network ConnecEvity

Depth of lake

Level of homogenizaEon

No

r o t c a f rmal

Channeling EC Zeeman again Generic picture for systems with minimizing dynamics Splizng Factors:

Strength of Feedback

Network ConnecEvity

Depth of lake

Level of homogenizaEon

Could offer insight into more policy opEons…

Channeling Christopher Zeeman Google: Sir Christopher Zeeman Christmas Lectures

THANK YOU!

Google: MPE 2013 MCRN Math Sir Christopher Zeeman, Christmas Lectures Thanks to: many friends & colleagues

MCRN